MELNIKOV ANALYSIS OF NUMERICALLY INDUCED CHAOS IN THE NONLINEAR SCHRODINGER-EQUATION

Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrodinger Equation (NLS) have been shown to be responsible for the generation of numerical instabilities and chaos. In this paper we unde rtake a dynamical systems type of approach to modeling the observed ir regular behavior of a conservative discretization of the NLS. Using he uristic Mel'nikov methods, the existence of a pair of isolated homocli nic orbits is indicated for the perturbed system. The structure of the persistent homoclinic orbits that are predicted by the Mel'nikov theo ry possesses the same features as the wave form observed numerically i n the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos. The Mel'nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agre ement with the numerical experiments. In a neighborhood of the persist ent homoclinic orbits, the existence of a horseshoe is investigated an d related with the onset of chaos in the numerical study.